A393681 - OEIS

%I #6 Mar 27 2026 00:13:41

%S 2,4,8,13,20,29,40,52,66,81,98,117,138,160,184,209,236,265,296,328,

%T 362,397,434,473,514,556,600,645,692,741,792,844,898,953,1010,1069,

%U 1130,1192,1256,1321,1388,1457,1528,1600,1674,1749,1826,1905,1986,2068,2152

%N Upper (1/2,1/3) midsequence of (n^2) and ((n+2)^2); see Comments.

%C Suppose that s = (s(n)) and t = (t(n)) are sequences of numbers and h > 0 and k > 0. The lower (h, k)-midsequence of s and t is floor(h*s + k*t); the upper (h, k)-midsequence of s and t is ceiling(h*s + k*t).

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,1,-2,1).

%F a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), with (a(0),...,a(5)) = (1, 3, 7, 12, 20).

%F G.f.: (-2 - 2*x^2 - x^3 - 2*x^4 - 2*x^5 - x^7)/((-1 + x)^3*(1 + x + x^2 + x^3 + x^4 + x^5)).

%e s = (n^2) = A000290 = (0, 1, 4, 9, 16, 25, 36, ...).

%e t = ((n+2)^2) = (4, 9, 16, 25, 36, 49, 64, 81, ...).

%e u(n) = (1, 3, 7, 12, 20, 28, 39, 51, 65, 80, 98, 116, 137, ...).

%e v(n) = (2, 4, 8, 13, 20, 29, 40, 52, 66, 81, 98, 117, 138, ...).

%t LinearRecurrence[{1,1,0,-1,-1,1}, {1, 3, 7, 12, 20, 28}, 30]    (* A393680 *)

%t LinearRecurrence[{2,-1,0,0,0,1,-2,1}, {2, 4, 8, 13, 20, 29}, 30]    (* A393681 *)

%Y Cf. A000290, A000578, A393680.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Mar 22 2026